The overall objective of this grant is to develop and apply statistical and quantitative methods in AIDS research. With over 29,00 AIDS cases reported to the Centers for Disease Control as of January 1987, the AIDS epidemic is becoming the most urgent public health problem in the United States. There are two specific objectives for this proposal. The first is to develop quantitative methods for projecting the course of the aids epidemic from the available data on AIDS incidence, human immunodeficiency virus (HIV) seroprevalence and knowledge of the transmission and natural history of HIV infection. The methods will be applied to obtain projections for each risk group in the United States. A subtopic will be the development of methods to evaluate the potential impact of transmission prevention programs and treatment intervention programs on numbers of future cases. The second objective is to develop statistical methods for studies of the epidemiology of AIDS. We will focus on the development of methods for estimating the distribution of the time to AIDS following infection. A number of important complexities are introduced because the date of infection is usually not know preciously, and because the infection rate changes over calendar time. The methods will be relevant to a number of important AIDS studies currently underway. We will apply the techniques to a hemophiliac cohort and the results of the calculations will be used in our work on projections of the course of the AIDS epidemic.